3.657 \(\int \frac{e^{-2 \coth ^{-1}(a x)}}{(c-a^2 c x^2)^{7/2}} \, dx\)

Optimal. Leaf size=98 \[ -\frac{8 x}{21 c^3 \sqrt{c-a^2 c x^2}}-\frac{4 x}{21 c^2 \left (c-a^2 c x^2\right )^{3/2}}-\frac{x}{7 c \left (c-a^2 c x^2\right )^{5/2}}+\frac{2 (1-a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}} \]

[Out]

(2*(1 - a*x))/(7*a*(c - a^2*c*x^2)^(7/2)) - x/(7*c*(c - a^2*c*x^2)^(5/2)) - (4*x)/(21*c^2*(c - a^2*c*x^2)^(3/2
)) - (8*x)/(21*c^3*Sqrt[c - a^2*c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.124215, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {6167, 6142, 653, 192, 191} \[ -\frac{8 x}{21 c^3 \sqrt{c-a^2 c x^2}}-\frac{4 x}{21 c^2 \left (c-a^2 c x^2\right )^{3/2}}-\frac{x}{7 c \left (c-a^2 c x^2\right )^{5/2}}+\frac{2 (1-a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^(7/2)),x]

[Out]

(2*(1 - a*x))/(7*a*(c - a^2*c*x^2)^(7/2)) - x/(7*c*(c - a^2*c*x^2)^(5/2)) - (4*x)/(21*c^2*(c - a^2*c*x^2)^(3/2
)) - (8*x)/(21*c^3*Sqrt[c - a^2*c*x^2])

Rule 6167

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

Rule 6142

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/c^(n/2), Int[(c + d*x^2)^(p
+ n/2)/(1 - a*x)^n, x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[a^2*c + d, 0] &&  !(IntegerQ[p] || GtQ[c, 0]) && I
LtQ[n/2, 0]

Rule 653

Int[((d_) + (e_.)*(x_))^2*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)*(a + c*x^2)^(p + 1))/(c*(
p + 1)), x] - Dist[(e^2*(p + 2))/(c*(p + 1)), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, p}, x] &&
EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && LtQ[p, -1]

Rule 192

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p + 1
], 0] && NeQ[p, -1]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{e^{-2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx &=-\int \frac{e^{-2 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx\\ &=-\left (c \int \frac{(1-a x)^2}{\left (c-a^2 c x^2\right )^{9/2}} \, dx\right )\\ &=\frac{2 (1-a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}}-\frac{5}{7} \int \frac{1}{\left (c-a^2 c x^2\right )^{7/2}} \, dx\\ &=\frac{2 (1-a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}}-\frac{x}{7 c \left (c-a^2 c x^2\right )^{5/2}}-\frac{4 \int \frac{1}{\left (c-a^2 c x^2\right )^{5/2}} \, dx}{7 c}\\ &=\frac{2 (1-a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}}-\frac{x}{7 c \left (c-a^2 c x^2\right )^{5/2}}-\frac{4 x}{21 c^2 \left (c-a^2 c x^2\right )^{3/2}}-\frac{8 \int \frac{1}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{21 c^2}\\ &=\frac{2 (1-a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}}-\frac{x}{7 c \left (c-a^2 c x^2\right )^{5/2}}-\frac{4 x}{21 c^2 \left (c-a^2 c x^2\right )^{3/2}}-\frac{8 x}{21 c^3 \sqrt{c-a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.061376, size = 96, normalized size = 0.98 \[ -\frac{\sqrt{1-a^2 x^2} \left (-8 a^5 x^5-16 a^4 x^4+4 a^3 x^3+24 a^2 x^2+9 a x-6\right )}{21 a c^3 (1-a x)^{3/2} (a x+1)^{7/2} \sqrt{c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^(7/2)),x]

[Out]

-(Sqrt[1 - a^2*x^2]*(-6 + 9*a*x + 24*a^2*x^2 + 4*a^3*x^3 - 16*a^4*x^4 - 8*a^5*x^5))/(21*a*c^3*(1 - a*x)^(3/2)*
(1 + a*x)^(7/2)*Sqrt[c - a^2*c*x^2])

________________________________________________________________________________________

Maple [A]  time = 0.043, size = 64, normalized size = 0.7 \begin{align*}{\frac{ \left ( ax-1 \right ) ^{2} \left ( 8\,{x}^{5}{a}^{5}+16\,{x}^{4}{a}^{4}-4\,{x}^{3}{a}^{3}-24\,{a}^{2}{x}^{2}-9\,ax+6 \right ) }{21\,a} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x+1)*(a*x-1)/(-a^2*c*x^2+c)^(7/2),x)

[Out]

1/21*(a*x-1)^2*(8*a^5*x^5+16*a^4*x^4-4*a^3*x^3-24*a^2*x^2-9*a*x+6)/a/(-a^2*c*x^2+c)^(7/2)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x-1)/(a*x+1)/(-a^2*c*x^2+c)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 3.03701, size = 250, normalized size = 2.55 \begin{align*} \frac{{\left (8 \, a^{5} x^{5} + 16 \, a^{4} x^{4} - 4 \, a^{3} x^{3} - 24 \, a^{2} x^{2} - 9 \, a x + 6\right )} \sqrt{-a^{2} c x^{2} + c}}{21 \,{\left (a^{7} c^{4} x^{6} + 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 2 \, a^{2} c^{4} x + a c^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x-1)/(a*x+1)/(-a^2*c*x^2+c)^(7/2),x, algorithm="fricas")

[Out]

1/21*(8*a^5*x^5 + 16*a^4*x^4 - 4*a^3*x^3 - 24*a^2*x^2 - 9*a*x + 6)*sqrt(-a^2*c*x^2 + c)/(a^7*c^4*x^6 + 2*a^6*c
^4*x^5 - a^5*c^4*x^4 - 4*a^4*c^4*x^3 - a^3*c^4*x^2 + 2*a^2*c^4*x + a*c^4)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a x - 1}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{7}{2}} \left (a x + 1\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x-1)/(a*x+1)/(-a**2*c*x**2+c)**(7/2),x)

[Out]

Integral((a*x - 1)/((-c*(a*x - 1)*(a*x + 1))**(7/2)*(a*x + 1)), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a x - 1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{7}{2}}{\left (a x + 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x-1)/(a*x+1)/(-a^2*c*x^2+c)^(7/2),x, algorithm="giac")

[Out]

integrate((a*x - 1)/((-a^2*c*x^2 + c)^(7/2)*(a*x + 1)), x)